This document discusses various parameters used to characterize two-port networks, including Z, Y, S, and ABCD parameters. It explains that Z parameters relate voltages and currents using an impedance matrix, while Y parameters use an admittance matrix. S parameters describe the scattering of waves at the ports in terms of reflection and transmission coefficients. The ABCD matrix relates voltages and currents at the ports and allows for easy cascading of networks. Measurements are typically made using a vector network analyzer to determine the S-parameter scattering matrix.

1. 2-Port Network Parameters
Mir Muhammad Lodro
Lecturer
Department of Electrical Engineering
Sukkur IBA

2. Contents
 Introduction
 Two Port Networks
 Z Parameters
 Y Parameters
 S Parameters
 Return Loss
 Insertion Loss
 Transmission (ABCD) Matrix
 VNA Calibration
 SOLT Structure
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3. Two Port Networks
 Linear networks can be completely characterized by parameters
measured at the network ports without knowing the content of the
networks.
 Networks can have any number of ports.
 Analysis of a 2-port network is sufficient to explain the theory and applies
to isolated signals (no crosstalk).
I1 I2
+ +
Port 2
Port 1
2 Port
V1 V2
-
Network -
 The ports can be characterized with many parameters (Z, Y, S,
ABDC). Each has a specific advantage.
 Each parameter set is related to 4 variables:
 2 independent variables for excitation
 2 dependent variables for response
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4. Z Parameters
Impedance Matrix: Z Parameters for 2-port Network
V1   Z11 Z12   I1 
V    Z  I  V   Z I 
 2   21 Z 22   2 
V1  Z11I1  Z12 I 2
V2  Z 21I1  Z 22 I 2
 Advantage: Z parameters are intuitive.
 Relates all ports to an impedance & is easy to calculate.
 Disadvantage: Requires open circuit voltage measurements, which are difficult
to make.
 Open circuit reflections inject noise into measurements.
 Open circuit capacitance is non-trivial at high frequencies.
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5. Y Parameters
Admittance Matrix: Y Parameters for 2-port Networks
 I1  Y11 Y12  V1  I   Y V 
 I   Y Y  V 
 2   21 22   2 
I1  Y11V1  Y12V2
I 2  Y21V1  Y22V2
 Advantage: Y parameters are also somewhat intuitive.
 Disadvantage: Requires short circuit voltage measurements, which are
difficult to make.
 Short circuit reflections inject noise into measurements.
 Short circuit inductance is non-trivial at high frequencies.
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6. Example- Z-parameters
ZA ZB
+ + V1  Z11I1  Z12 I 2
I1 I2
V2  Z 21I1  Z 22 I 2
Port 1
Port 2
V1 ZC V2
- -
V1 V1 V1 I 2 ZC
Z11 
I1

V1
 Z A  ZC Z12    ZC
I 20
Z A  ZC
I2 I10
I2
V2 I1 Z C V2 V2
Z 21    ZC Z 22    Z B  ZC
I2 V2
I1 I 2 0
I1 I10
Z B  ZC
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7. Frequency Domain: Vector Network Analyzer
(VNA)
 VNA offers a means to characterize circuit elements as a function of
frequency.
 VNA is a microwave based instrument that provides the ability to
understand frequency dependent effects.
 The input signal is a frequency swept sinusoid.
 Characterizes the network by observing transmitted and reflected power
waves.
 Voltage and current are difficult to measure directly.
 It is also difficult to implement open & short circuit loads at high frequency.
 Matched load is a unique, repeatable termination, and is insensitive to length,
making measurement easier.
 Incident and reflected waves the key measures.
 We characterize the device under test using S parameters.
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8. S Parameters
a1 a2
+ +
Port 2
Port 1
2 Port
V1 b1 b2 V2
-
Network -
 We wish to characterize the network by observing transmitted and
reflected power waves.
 ai represents the square root of the power wave injected into port i.
V1
ai  P  P V2
R R
 bi represents the square root of the power wave injected into port j.
V j
bj 
R
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9. S Parameters
a1 a2
 We can use a set of linear equations
+ +
to describe the behavior of the
Port 2
Port 1
2 Port
network in terms of the injected and V1 b1 b2 V2
reflected power waves. -
Network -
b1  S11a1  S12a2
bj
power measured at port j
b2  S 21a1  S 22a2 Sij  
ai power measured at port i
 For the 2 port case:
 b1   S11 S12   a1 
b   S S 22  a2 
 2   21  
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10. Scattering Matrix – Return Loss
 S11, the return loss, is a measure of
the power returned to the source.
 When there is no reflection from the
load, or the line length is zero, S11 is
equal to the reflection coefficient.
V1
b Z0 V1 Vreflected Z  50
S11  1      0  0
a1 a 2 0
V1 V1 Vincident Z 0  50
Z0
In general:
bi
Sii   0
ai a j 0 , j 0
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11. Scattering Matrix – Return Loss
 When there is a reflection from the load, S11 will be composed of
multiple reflections due to standing waves.
 Use input impedance to calculate S11 when the line is not
perfectly terminated.
1   ( z  0)
Z in  Z ( z  0)  Z o
1   ( z  0)
RS = 50 S11 for a transmission line will exhibit
periodic effects due to the standing waves.
Zin  If the network is driven with a 50 source, S11
is calculated as follows:
Z in  50
S11   v 
Z in  50
In this case S11 will be maximum when Zin is real. An imaginary component implies a
phase difference between Vinc and Vref. No phase difference means they are perfectly
aligned and will constructively add.
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12. Scattering Matrix – Insertion Loss
a1 a2
+ +
Port 2
Port 1
Z0
2 Port
V1 b1 b2 Z0 V2
-
Network -
 When power is injected into Port 1 and measured at Port 2, the power
ratio reduces to a voltage ratio:

V2
b2 Z o V2  Vtransmitted
S 21      
a1 a 20 V1 V1 Vincident
Zo
 S21, the insertion loss, is a measure of the power
transmitted from port 1 to port 2.
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13. S Parameters
Sij = Gij is the reflection coefficient of the ith
port if i=j with all other ports matched
Sij = Tij is the forward transmission coefficient
of the ith port if I>j with all other ports
matched
Sij = Tij is the reverse transmission coefficient b  S a
of the ith port if I<j with all other ports
matched
Vi 
bi
Sij  Sij 
bi

Z 0i
aj aj V j
a k 0 , k  j a k 0 , k  j
Z0 j
Vk 0 ,k  j

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14. Comments on Losses
 True losses come from physical energy losses.
 Ohmic (i.e. skin effect)
 Field dampening effects (loss tangent)
 Radiation (EMI)
 Insertion and return losses include other effects, such as
impedance discontinuities and resonance, which are not true
losses.
 Loss free networks can still exhibit significant insertion and
return losses due to impedance discontinuities.
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15. Reflection Coefficients
 Reflection coefficient at the load:
Z L  Z0
L 
Z L  Z0
 Reflection coefficient at the source:
Z S  Z0
S 
Z S  Z0
 Input reflection coefficient:
S12 S 21 L S12  L
2
in  S11   S11  Assuming S12 = S21 and S11 = S22.
1  S 22  L 1  S11 L
 Output reflection coefficient:
S12 S 21 S
 out  S 22 
1 S11 S
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16. Transmission Line Z0 Measurements
 Impedance vs. frequency
 Recall
1  e 2 jl
Z in  Z 0
1  e 2 jl
 Zin vs f will be a function of delay () and ZL.
 We can use Zin equations for open and short circuited lossy
transmission.
Z in,open  Z 0 tanh l 
Z 0  Z in, shortZ in,open
Z in,short  Z 0 coth l 
Using the equation for Zin, rin, and Z0,
we can find the impedance.
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17. Advantages/Disadvantages of S Parameters
Advantages:
 Ease of measurement: It is much easier to measure
power at high frequencies than open/short current and
voltage.
Disadvantages:
 They are more difficult to understand and it is more
difficult to interpret measurements.
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18. Transmission (ABCD) Matrix
 The transmission matrix describes the network in terms of both voltage
and current waves (analagous to a Thévinin Equivalent).
V1  AV2  BI 2 I1 I2
I1  CV2  DI 2 + +
Port 1
Port 2
V1
2 Port V2
Network
V1 A B V2 - -

I1 C D I2
 The coefficients can be defined using superposition:
V1 V1
A B
V2 I 2 0 I2 V2  0
I1
C D
I1
V2 I 2 0 I2 V2  0
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19. Transmission (ABCD) Matrix
 Since the ABCD matrix represents the ports in terms of currents and
voltages, it is well suited for cascading elements.
I1 I2 I3
+ A B A B +
V1 V2 V3
C D1 C D2
- -
I1
 The matrices can be mathematically cascaded by multiplication:
V1 A B V2
 
I1 C D 1 I2 V1 A B A B V3
  
V2 A B V3 I1 C D 1 C D 2 I 3
 
I2 C D 2 I3
 This is the best way to cascade elements in the frequency domain.
 It is accurate, intuitive and simple to use.
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20. Converting to and from the S-Matrix
 The S-parameters can be measured with a VNA, and
converted back and forth into ABCD, the Matrix
 Allows conversion into a more intuitive matrix
 Allows conversion to ABCD for cascading
 ABCD matrix can be directly related to several useful
circuit topologies
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21. Advantages/Disadvantages of ABCD Matrix
Advantages:
 The ABCD matrix is intuitive: it describes all ports with voltages and
currents.
 Allows easy cascading of networks.
 Easy conversion to and from S-parameters.
 Easy to relate to common circuit topologies.
Disadvantages:
 Difficult to directly measure: Must convert from measured
scattering matrix.
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22. VNA Calibration
 Proper calibration is critical!!!
 There are two basic calibration methods
 Short, Open, Load and Thru (SOLT)
 Calibrated to known standard( Ex: 50)
 Measurement plane at probe tip
 Thru, Reflect, Line(TRL)
 Calibrated to line Z0
 Helps create matched port condition.
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23. SOLT Calibration Structures
OPEN SHORT
S S
Signal
G G
Ground
LOAD THRU
S S S
G G G
Calibration Substrate
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24. Thanks
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